Average Error: 34.4 → 10.1
Time: 16.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.635925081143504476780080161813975782827 \cdot 10^{-66}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.206904744652339671334892722279467095293 \cdot 10^{101}:\\ \;\;\;\;\frac{-\left(\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -8.635925081143504476780080161813975782827 \cdot 10^{-66}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.206904744652339671334892722279467095293 \cdot 10^{101}:\\
\;\;\;\;\frac{-\left(\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b\right)}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r3605935 = b;
        double r3605936 = -r3605935;
        double r3605937 = r3605935 * r3605935;
        double r3605938 = 4.0;
        double r3605939 = a;
        double r3605940 = c;
        double r3605941 = r3605939 * r3605940;
        double r3605942 = r3605938 * r3605941;
        double r3605943 = r3605937 - r3605942;
        double r3605944 = sqrt(r3605943);
        double r3605945 = r3605936 - r3605944;
        double r3605946 = 2.0;
        double r3605947 = r3605946 * r3605939;
        double r3605948 = r3605945 / r3605947;
        return r3605948;
}
↓
double f(double a, double b, double c) {
        double r3605949 = b;
        double r3605950 = -8.635925081143504e-66;
        bool r3605951 = r3605949 <= r3605950;
        double r3605952 = -1.0;
        double r3605953 = c;
        double r3605954 = r3605953 / r3605949;
        double r3605955 = r3605952 * r3605954;
        double r3605956 = 3.2069047446523397e+101;
        bool r3605957 = r3605949 <= r3605956;
        double r3605958 = r3605949 * r3605949;
        double r3605959 = 4.0;
        double r3605960 = r3605953 * r3605959;
        double r3605961 = a;
        double r3605962 = r3605960 * r3605961;
        double r3605963 = r3605958 - r3605962;
        double r3605964 = sqrt(r3605963);
        double r3605965 = r3605964 + r3605949;
        double r3605966 = -r3605965;
        double r3605967 = 2.0;
        double r3605968 = r3605967 * r3605961;
        double r3605969 = r3605966 / r3605968;
        double r3605970 = 1.0;
        double r3605971 = r3605949 / r3605961;
        double r3605972 = r3605954 - r3605971;
        double r3605973 = r3605970 * r3605972;
        double r3605974 = r3605957 ? r3605969 : r3605973;
        double r3605975 = r3605951 ? r3605955 : r3605974;
        return r3605975;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.4
Target20.9
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.635925081143504e-66

    1. Initial program 53.4

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around -inf 8.4

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    if -8.635925081143504e-66 < b < 3.2069047446523397e+101

    1. Initial program 13.4

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv13.5

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}}\]
    4. Using strategy rm

    5. Applied associate-*r/13.4

      \[\leadsto \color{blue}{\frac{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot 1}{2 \cdot a}}\]

    6. Simplified13.5

      \[\leadsto \frac{\color{blue}{-\left(\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b\right)}}{2 \cdot a}\]

    if 3.2069047446523397e+101 < b

    1. Initial program 46.8

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around inf 4.4

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    3. Simplified4.4

      \[\leadsto \color{blue}{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.635925081143504476780080161813975782827 \cdot 10^{-66}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.206904744652339671334892722279467095293 \cdot 10^{101}:\\ \;\;\;\;\frac{-\left(\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))